Find the distance between $(0, 0)$ and $(3, 4)$.

How we read your problem

Find the straight-line (Euclidean) distance between two points in the plane.

Step-by-step solution

1. 1

   Label the points

   $$(x_1, y_1) = (0, 0),\quad (x_2, y_2) = (3, 4)$$
2. 2

   Find the horizontal and vertical differences

   $$\Delta x = 3,\quad \Delta y = 4$$
3. 3

   Square the differences and add them

   $$(3)^2 + (4)^2 = 9 + 16 = 25$$
4. 4

   Take the square root

   $$d = \sqrt{25} = 5$$
5. 5

   Midpoint (average of the coordinates)

   $$M = \left(\frac{0 + 3}{2}, \frac{0 + 4}{2}\right) = (1.5, 2)$$

Formulas used

• Distance formula

  $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

• Midpoint formula

  $$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

The distance formula is the Pythagorean theorem applied to the right triangle whose legs are the horizontal and vertical differences between the points.

Verification

• d² = 25 matches Δx² + Δy² = 25